Assume $\displaystyle f: [0, 2\pi] \rightarrow \mathbb{R}$ is Lipschitz continuous. Prove that exists constans $\displaystyle C>0$ that for every $\displaystyle k=1,2...$ there is:
$\displaystyle \int^{2\pi}_{0} f(x) \sin (kx) dx \leq \frac{C}{k}$.
Assume $\displaystyle f: [0, 2\pi] \rightarrow \mathbb{R}$ is Lipschitz continuous. Prove that exists constans $\displaystyle C>0$ that for every $\displaystyle k=1,2...$ there is:
$\displaystyle \int^{2\pi}_{0} f(x) \sin (kx) dx \leq \frac{C}{k}$.